Are they hakurei turnips? My local farmer grew them, and I tried them for the first time 2 years ago. Oh my. I love turnips, but these are the best things you've ever eaten in all your life. This year, my mom and my dh are growing them in their gardens. And, I think I've eaten every single one. I like them raw, peeled, and cold from the fridge, with a little salt.
But, they are also really good grilled, too.
As for the greens--I grew up eating turnip greens only. We didn't even wait for them to make roots. I didn't realize that people ate the roots regularly until I was an adult.
Cook them like any other green--they are one of my favorites.